(; c. 953 – c. 1029) was a 10th-century

Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples ** Persian language, an Iranian language of the ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

and

engineer Engineers, as practitioners of engineering, are professionals who Invention, invent, design, build, maintain and test machines, complex systems, structures, gadgets and materials. They aim to fulfill functional objectives and requirements while ...

who flourished at

Baghdad Baghdad ( or ; , ) is the capital and List of largest cities of Iraq, largest city of Iraq, located along the Tigris in the central part of the country. With a population exceeding 7 million, it ranks among the List of largest cities in the A ...

. He was born in

Karaj Karaj (; ) is a List of cities in Iran by province, city in the Central District (Karaj County), Central District of Karaj County, Alborz province, Alborz province, Iran, serving as capital of the province, the county, and the district. Earl ...

, a city near

Tehran Tehran (; , ''Tehrân'') is the capital and largest city of Iran. It is the capital of Tehran province, and the administrative center for Tehran County and its Central District (Tehran County), Central District. With a population of around 9. ...

. His three principal surviving works are mathematical: ''Al-Badi' fi'l-hisab'' (''Wonderful on calculation''), ''Al-Fakhri fi'l-jabr wa'l-muqabala'' (''Glorious on algebra''), and ''Al-Kafi fi'l-hisab'' (''Sufficient on calculation'').

Work

Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by

Diophantus Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...

) but most regard him as more original, in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book ''al-fakhri fi al-jabr wa al-muqabala'', which survives from the medieval era in at least four copies. He expounded the basic principles of hydrology and this book reveals his profound knowledge of this science and has been described as the oldest extant text in this field. He systematically studied the algebra of exponents, and was the first to define the rules for

monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...

s like x, x², x³ and their reciprocals in the cases of multiplication and division. However, since for example the product of a square and a cube would be expressed, in words rather than in numbers, as a square-cube, the numerical property of adding exponents was not clear. His work on

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

and

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials; though he was restricted to dividing polynomials by monomials. F. Woepcke was the first historian to realise the importance of al-Karaji's work and later historians mostly agree with his interpretation. He praised Al-Karaji for being the first who introduced the theory of algebraic calculus. Al-Karaji gave an early formulation of the

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

s and the first description of

Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...

. He is also presumed to have discovered the binomial theorem. In a now lost work known only from subsequent quotation by al-Samaw'al, Al-Karaji introduced the idea of argument by

mathematical induction Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a ...

. As Katz says

Notes

References and external links

* * * J. Christianidis. ''Classics in the History of Greek Mathematics'', p. 260 * Carl R. Seaquist, Padmanabhan Seshaiyer, and Dianne Crowley
"Calculation across Cultures and History"
(''Texas College Mathematics Journal'' 1:1, 2005; pp 15–31) DF* Matthew Hubbard and Tom Roby
"The History of the Binomial Coefficients in the Middle East"
from "Pascal's Triangle from Top to Bottom") * Fuat Sezgin. ''Geschichte des arabischen Schrifttums'' (1974, Leiden: E. J. Brill) * James J. Tattersall. ''Elementary Number Theory in Nine Chapters'', p. 32 *

Mariusz Wodzicki Mariusz Wodzicki (Polish: ; born 1956) is a Polish mathematician and professor of mathematics at the University of California, Berkeley, whose works primarily focus on analysis, algebraic k-theory, noncommutative geometry, and algebraic geometry. ...

"Early History of Algebra: a Sketch"
(''Math'' 160, Fall 2005) DF
"al-Karaji"
— ''Encyclopædia Britannica'' Online (4 April 2006)
Extrait du ''Fakhri'', traité d'Algèbre par Abou Bekr Mohammed Ben Alhaçan Alkarkhi
presented with commentary by F. Woepcke, year 1853. {{DEFAULTSORT:Karaji 950s births 1029 deaths 10th-century Iranian mathematicians 11th-century Iranian mathematicians People from Karaj

Work

See also

Notes

References and external links